Cycles Realistic Lighting & Emission material
Strength of the emitted light. For point and area lamps, the unit is Watts. For materials, a value of 1.0 will ensure that the object in the image has the exact same color as the Color input, i.e. make it ‘shadeless’.


This is junk documentation, written by someone that clearly doesn't understand scene referred values.

A material albedo value of 1.0 means that 100% of the light is reflected back. This does not result in the exact same value as the colour input, which depends on the transfer function and other potential sweeteners in the transform from the scene referred to the display referred domain.

While I can't answer the wattage question any better than the other answers on this site already well written, I can speak to exposure. Exposure is always a creative choice. Glowing eyes will likely always be less glowing than a screaming ball of fire in the sky that is gargantuan. That is a creative issue that could be negotiated with creative compositing or creative exposure within your scene.


  Does this mean that Emission material strength doesn't change regardless of the lighting?


Emissions are scene referred. That is, the effective range of a value is some potentially infinitely small value to some potentially infinitely large value. If you set an emission to 146281.8, it is that value. How that value ends up in your image is a complex byproduct of the aforementioned transform from the scene referred to the display referred domain. That is, think of the transform as a sort of film stock. Various film stocks may have different colour responses, different dynamic ranges, and a few other aesthetic elements that will take a physical scene quantity of light and convert them into an aesthetic output.


  Do materials need to be changed according to the scene (lighting, exposure), or is the simulation realistic and materials only need to be configured for their realistic physical properties?


Starting with physically plausible values is a great entry point. Ultimately however, materials should serve the creative needs.


  Is it normal that realistic light values completely destroy the scene when using default Exposure, and it needs to be dialed way down?


Is it normal for a camera to take an overexposed photo when set to manual? Same idea applies here; exposure is a creative choice, as is balancing lighting in a scene, either starting with realistically plausible values or creatively lighting a piece with artificial sources.


  Is there a way to calculate realistic, physically correct emission for materials? (Ex. the emission of an LED lamp on a monitor)


Yes. This is a duplicate question with more than a few good answers on this site.